On the solution of a mixed nonlinear integral equation

In this paper, we consider a mixed nonlinear integral equation of the second kind in position and time. The existence of a unique solution of this equation is discussed and proved. A numerical method is used to obtain a system of Harmmerstein integral equations of the second kind in position. Then the modified Toeplitz matrix method, as a numerical method, is used to obtain a nonlinear algebraic system. Many important theorems related to the existence and uniqueness solution to the produced nonlinear algebraic system are derived. The rate of convergence of the total error is discussed. Finally, numerical examples when the kernel of position takes a logarithmic and Carleman forms, are presented and the error estimate, in each case, is calculated.     

The Hijazi inequality on conformally parabolic manifolds

We prove the Hijazi inequality, an estimate for Dirac eigenvalues, for complete manifolds of finite volume. Under some additional assumptions on the dimension and the scalar curvature, this inequality is also valid for elements of the essential spectrum. This allows to prove the conformal version of the Hijazi inequality on conformally parabolic manifolds if the spin analog to the Yamabe invariant is positive.     

OH�� Radical Production Induces Direct Enhancement of the Amylo?d �� Protein/Chondroitin Sulfate Binding: Inhibition by Potentially Radical Scavengers, a Biochromatographic and Thermodynamic Model

??-Amyloid (A??) is a major component of the senile plaques characteristic of Alzheimer disease (AD). Chondroitin sulfate (CS) and glycoaminoglycan (GAG) are also localized throughout the senile plaques in AD. In previous studies, the interaction of the A?? protein with CS immobilized on a chromatographic support and the role of aluminum and copper cations was studied using a molecular chromatographic approach [1, 2]. Here, we demonstrated the direct implication of OH· radical formation on this binding via a novel analytical procedure. The binding of A?? amyloid on CS was accompanied by an OH· radical uptake. The A?¨CCS complex was stabilized by the OH· radical via the creation of about one to two hydrogen bonds. The addition in the medium of a radical scavenger allowed decreasing the A??/CS association and thus confirmed the positive role of these compounds in amyloidosis.     

Lanthanide-mediated supramolecular cages and host-guest interactions

The structure and thermodynamic properties of lanthanide complexes with a new tripodal ligand L2 have been elucidated using different physicochemical methods. At stoichiometric ratios, the tetrahedral three-dimensional complexes with lanthanide cations are formed in acetonitrile with good stabilities. Despite minor structural changes comparing to previously investigated tripodal ligands, the resulting assembly exhibits different features revealed with the crystal structure of [Eu(4)L2(4)](OH)(ClO(4))(11) (orthorhombic, Pbcn). Interestingly, the highly charged edifice contains an inner cage encapsulating a perchlorate anion. Such lanthanide mediated cage-like assemblies are rare, and may be of interest for different sensing applications. Indeed, the anionic guest can be exchanged with different anions. The related host-guest equilibria were investigated with NMR techniques. Various aspects of these reactions are qualitatively discussed.     

Solid‐particle erosion in the tube end of the tube sheet of a shell‐and‐tube heat exchanger

Erosion is one of the major problems in many industrial processes, and in particular, in heat exchangers. The effects of flow velocity and sand particle size on the rate of erosion in a typical shell‐and‐tube heat exchanger were investigated numerically using the Lagrangian particle‐tracking method. Erosion and penetration rates were obtained for sand particles of diameters ranging from 10 to 500 µm and for inlet flow velocities ranging from 0.197 to 2.95 m/s. A flow visualization experiment was conducted with the objective of verifying the accuracy of the continuous phase calculation procedure. Comparison with available experimental data of penetration rates was also conducted. These comparisons resulted in a good agreement. The results show that the location and number of eroded tubes depend mainly on the particle size and velocity magnitude at the header inlet. The rate of erosion depends exponentially on the velocity. The particle size shows negligible effect on the erosion rate at high velocity values and the large‐size particles show less erosion rates compared to the small‐size particles at low values of inlet flow velocities. The results indicated that the erosion and penetration rates are insignificant at the lower end of the velocity range. However, these rates were found to increase continuously with the increase of the inlet flow velocity for all particle sizes. The particle size creating the highest erosion rate was found to depend on the flow velocity range. Copyright © 2005 John Wiley & Sons, Ltd.     

Spatiotemporal and statistical symmetries

The notion of symmetries, either statistical or deterministic, can be useful for the characterization of complex systems and their bifurcations. In this paper, we investigate the connection between the (microscopic) spatiotemporal symmetries of a space-time functionu(x, t), on the one hand, and the (macroscopic) symmetries of statistical quantities such as the spatial (resp. temporal) two-point correlations and the spatial (resp. temporal) average, on the other hand. We show, how, under certain conditions, these symmetries are related to the symmetries of the orbits described byu(x, t) in the characteristic (phase) spaces. We also determine the largest group of spatiotemporal symmetries (in the sense introduced in our earlier work) satisfied by a given space-time functionu(x, t) and indicate how to extract the subgroups of point symmetries, namely those directly implemented on the space and time variables. Conversely, we determine all the functions invariant by a given space-time symmetry group. Finally, we illustrate all the previous points with specific examples.     

Numerical investigation of natural convection inside an inclined parallel‐walled channel

Steady two‐dimensional natural convection in an inclined parallel‐walled channel was investigated numerically. The full elliptic forms of conservation equations were solved together and the velocity vectors, temperature contours and local and average Nusselt number distribution were obtained. The comparisons of local and average Nusselt number with published experimental and numerical results indicate very good agreement. Results are presented for a single aspect ratio, L/b=24, over the range of Rayleigh number of 3–1000 and angle of inclination 0–90°. The results indicate that the overall channel average Nusselt number is reduced as the inclination angle is increased. Significant reductions in the overall Nusselt number are exhibited at high angle of channel inclination. Copyright © 2005 John Wiley & Sons, Ltd.     

Spectroscopic studies of structural changes in irradiated LDPE and its blends

The low temperature glass transition occurring at low temperature in polyethylene samples was studied using Fourier transform infrared, FT‐Raman spectroscopic techniques, which are known to be sensitive to any structural changes in the investigated samples. Anomalous behavior was obtained in the temperature range (98 K → 383 K) indicating the existence of phase transition occurring at about 98 K and disappearing at about 323 K while heating for unirradiated virgin low density polyethylene (VPE) sample with or without crosslinking agent (trimethylol propane triacrylate, TMPTA). Addition of TMPTA monomer caused severe changes in the temperature dependence of both unirradiated and irradiated polyethylene samples. The results given indicated the occurrence of abnormal temperature dependence, which is thought to be related with low temperature structural change resulting from local crosslinking in scrapped polyethylene (SPE) sample already suffering from high degree of crosslinking. Differential scanning calorimetry (DSC) thermograms confirmed the observed new phase transition observed in both unirradiated and irradiated TMPTA loaded VPE/SPE (50/50 wt %) blend. Irradiation, addition of TMPTA as a crosslinking agent, and blending VPE with SPE were found to affect remarkably the variation of the vibration spectrum of LDPE and consequently the resulting structural changes. The experimental results obtained were discussed and correlated with reorientational disorders of the molecular segments (local crosslinking). © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 850–859, 2007     

Response behavior of sodium-selective electrodes modified by surface attachment of the anticoagulant polysaccharides heparin and chondroitin sulfate

Two different polysaccharides with anticoagulant activities, heparin and chondroitin sulfate, were used to modify the surface of sodium-selective electrodes based on asymmetric cellulose triacetate (CTA) membranes. The membranes were formulated with sodium ionophore X, anionic additive, and o-nitrophenyl octyl ether. The response behavior of the surface-modified sodium electrodes was compared with that of control CTA, as well as poly(vinyl chloride) (PVC)-based sodium-selective electrodes. It was found that the selectivity coefficients obtained with the surface modified CTA membrane electrodes were slightly higher than those of the control, but in the case of heparin-modified electrodes they still met the requirements for analysis of sodium in physiological fluids within an error of <1%; the corresponding error for chondroitin sulfate-modified electrodes was also <1% except for the case of potassium ion in which the error was 1.3%. Likewise, it was found that other response characteristics, such as detection limit, linear range, slope of the response plot, selectivity pattern, and response time were comparable in both the control and the polysaccharide-modified electrodes. Therefore, the surface modification does not significantly alter the response behavior of the sensors.